arXiv math/05106652005-10math.GRmath.GT

Averaged Dehn Functions for Nilpotent Groups

Robert Young

备注：: 15 pages, 1 figure; many corrections/clarifications, expanded proof of main theorem, added Proposition 4. To appear in Topology

英文摘要：

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl^α$ for $α>2$ then it satisfies the averaged isoperimetric inequality $δ^{\text{avg}}(l)<C'l^{α/2}$. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Scaled relators and Dehn functions for nilpotent groups

A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. Using this method, we give new proofs of some results on Dehn functions of nilpotent groups, prove theorems on central powers and certain quotients of nilpotent groups, and construct the first example of …

2006-02-07math.GRmath.GT

Nilpotent groups without exactly polynomial Dehn function

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.

2010-04-16math.DGmath.GR

具有Dehn函数$n^2 \log n$的幂零群

Nilpotent groups with Dehn function $n^2 \log n$

KU Leuven（荷语鲁汶大学）

该研究构造了具有Dehn函数$n^2 \log n$的2步Carnot群，改进了Wenger的下界，得到首个幂零群中不按$n^\alpha$（$\alpha \geq 1$）增长的Dehn函数的精确计算结果。

2026-08-26math.GRmath.MG

关于幂零群中心积的Dehn函数

On the Dehn functions of central products of nilpotent groups

University of Warwick（华威大学）；University of Luxembourg（卢森堡大学）；Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，联合研究单位8524-保罗·潘勒瓦实验室）

本文确定了多类幂零李群中心积的Dehn函数，证实了相关猜想，并构造了无限族群使Cornulier的界渐近最优。

2026-09-28math.GRmath.GTmath.MG

Windings of planar random walks and averaged Dehn function

We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

2009-12-07math.GRmath.GTmath.PR

A coarse embedding theorem for homological filling functions

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension $2$, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension $2$, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.

2020-11-18math.GRmath.GT

更多相关论文收起更多论文

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…

2018-09-17math.GRmath.GT

具有不同Dehn函数的锥等价幂零群

Cone-equivalent nilpotent groups with different Dehn functions

本研究构造了一族k步幂零李群，其与对应Carnot分次群渐近锥双李普希茨等价但Dehn函数增长阶不同，揭示了三类新现象并回答了相关领域的公开问题。

2023-08-23math.DGmath.GRmath.GT

On a generalization of Dehn's algorithm

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. W…

2008-01-15math.GRmath.GT

Filling inequalities for nilpotent groups

We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with arbitrarily large nilpotency class that have euclidean n-dimensional filling volume functions, and a proof of part of a conjecture of Gromov on the higher-order filling functions of the higher-dimensional Heisenberg groups.

2011-03-22math.GRmath.GT

弱交换性、虚拟幂零群与 Dehn 函数

Weak commutativity, virtually nilpotent groups, and Dehn functions

本文研究弱交换构造 $\mathfrak{X}(G)$，通过分析正规交换子群模结构，证明其保持虚拟幂零、Engel 与增长类型，并刻画字问题及 Dehn 函数下界。

2023-08-20math.GR

Higher divergence for nilpotent groups

The higher divergence of a metric space describes its isoperimetric behaviour at infinity. It is closely related to the higher-dimensional Dehn functions, but has more requirements to the fillings. We prove that these additional requirements do not have an essential impact for many nilpotent Lie groups. As a corollary, we obtain the higher divergence of the Heisenberg groups in all dimensions.

2018-07-26math.GRmath.MG

Combing nilpotent and polycyclic groups

A combing is a set of normal forms for a finitely generated group. This article investigates the language-theoretic and geometric properties of combings for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are also all 2 or 3-generated class 2 nilpotent groups, and groups in certain families of nilpotent groups, e.g. the finitely generated Heisenberg groups, groups of unipotent matrices over the integers and the free class 2 nilpotent groups. Further it is shown that any polycyclic-by-finite…

1999-01-21math.GR

Filling functions

Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes. Examples are the Dehn (or isoperimetric) function, the filling length function and the intrinsic diameter (or isodiametric) function. We discuss filling functions from geometric, combinatorial and computational points of view, we survey their interrelationships, and we sketch their roles in the studies of nilpotent groups, hyperbolic groups and asymptotic cones.…

2006-06-23math.GR